Problem: Let N be a positive odd number.

There are N coins, numbered 1, 2, \ldots, N. For each i (1 \leq i \leq N), when Coin i is tossed, it comes up heads with probability p_i and tails with probability 1 - p_i.

Taro has tossed all the N coins. Find the probability of having more heads than tails.

Constraints

* N is an odd number.
* 1 \leq N \leq 2999
* p_i is a real number and has two decimal places.
* 0 < p_i < 1

Input

Input is given from Standard Input in the following format:


N
p_1 p_2 \ldots p_N


Output

Print the probability of having more heads than tails. The output is considered correct when the absolute error is not greater than 10^{-9}.

Examples

Input

3
0.30 0.60 0.80


Output

0.612


Input

1
0.50


Output

0.5


Input

5
0.42 0.01 0.42 0.99 0.42


Output

0.3821815872

Solution (in JAVA):

Solution:
import java.util.*;
public class Main {
    public static void main(String args[]) {
      Scanner sc=new Scanner(System.in);
      int n=sc.nextInt();
      double ar[]=new double[n];
      for(int i=0;i<n;i++){
          ar[i]=sc.nextDouble();
      }
      int num = (int)Math.ceil((n+1)/2);
      double dp[][]=new double[n+1][n+1];
      for(double arr[]:dp){
          Arrays.fill(arr, -1.0);
      }
      double sum=0.0;
      for(int i=num;i<=n;i++){
          
          sum+=findCoins(dp, i, n, ar,0);
      }
      System.out.println(sum);
    }
    public static double findCoins(double[][] dp, int heads, int coins, double ar[], int st){
        
        if(heads > coins){
            return dp[coins][heads]=0.0;
        }
        if(coins==0){
            return  dp[coins][heads]=1.0;
           
        }
        if(dp[coins][heads]!=-1.0){
            return dp[coins][heads];
        }
        if(heads>0) {
            dp[coins][heads]=ar[st]*findCoins(dp,heads-1, coins-1,ar,st+1)+(1-ar[st])*findCoins(dp,heads, coins-1,ar,st+1);
        }
        else{
            dp[coins][heads] = (1-ar[st])*findCoins(dp,heads, coins-1,ar,st+1);
        }
        return dp[coins][heads];
    }
    
}